There are other topological properties of \(\Sigma \)  that result in \(\operatorname{Map} (\Sigma )\)  having a non-trivial geometry.
A subsurface \(R\)  of \(\Sigma \)  is called non-displaceable if for every \(f \in \operatorname{Map} (\Sigma )\) , \(f(R)\)  intersects \(R\) .
In [1]}, it was shown that \(\operatorname{Map} (\Sigma )\)  acts non-trivially on an infinite diameter hyperbolic space if and
only if \(\Sigma \)  contains a non-displaceable subsurface. Using the classification of coarsely bounded mapping
class groups given in [2]}, we show:
